Unit 8

“Quadrilaterals”

Academic Geometry

Spring 2014

Name_____________________________ Teacher__________________ Period______

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Unit 8 at a glance

“Quadrilaterals”

This unit focuses on revisiting prior knowledge of polygons and extends to formulate, test,

and apply conjectures about quadrilaterals. Students will be able to identify

quadrilaterals by the given properties and apply the properties to solve both purely

mathematical and real world situations.

Essential Questions

How are polygons related? What properties do they share?

What are the differences and similarities between the types of quadrilaterals (square,

rectangle, rhombus, parallelogram, trapezoid, and kite)? And how does knowing this

help me?

In Unit 8, students will…

Apply the terms convex, concave, n-gon, equilateral, equiangular, and regular to

describe polygons when solving problems involving area, perimeter and circumference

in both real-world and purely mathematical situations;

Formulate, test, and apply conjectures (based on explorations and concrete models)

involving the sum of the measures of interior and exterior angles of convex polygons

and the measures of each interior and exterior angle of a regular polygon to solve

problems in both real-world and purely mathematical situations;

Formulate, test, and apply conjectures (based on explorations and concrete models)

involving the properties of parallelogram (including angle and side measure

relationships) to solve problems in both real-world and purely mathematical

situations;

Formulate, test, and apply conjectures (based on explorations and concrete models)

involving the conditions that ensure a quadrilateral is a parallelogram including

opposite side, opposite angle and diagonal relationships and solve problems in both

real-world and purely mathematical situations requiring axiomatic and coordinate

approaches;

Formulate, test, and apply conjectures (based on explorations and concrete models)

involving the properties of rhombuses, rectangles and squares to solve problems in

both real-world and purely mathematical situations;

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Formulate, test, and apply conjectures (based on explorations and concrete models)

involving the properties of trapezoids and kites to solve problems in both real-world

and purely mathematical situations requiring axiomatic or coordinate geometry

approaches;

Compare and contrast quadrilaterals (parallelograms, rhombuses, rectangles, squares,

trapezoids, kites) and their properties to identify them and to solve problems in both

real-world and purely mathematical situations.

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Vocabulary

concave polygons

convex polygons

diagonal

equiangular

exterior angles

interior angles

isosceles trapezoid

kite

midsegment

n-gons

opposite angle

parallelogram

quadrilateral

rectangle

regular polygons

rhombus

square

trapezoid

vertices

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Meet the Quadrilateral Family

*Kite I have:

- 2 sets of consecutive congruent sides

- only one pair of opposite angles congruent

- diagonals perpendicular

*Parallelogram

I have:

- 2 sets of parallel sides

- 2 sets of congruent sides

- opposite angles congruent

- consecutive angles supplementary

- diagonals bisect each other

- diagonals form 2 congruent triangles

*Quadrilateral

I have exactly four sides.

The sum of my interior angles is 3600.

*Rhombus

I have all of the properties of the parallelogram PLUS

- 4 congruent sides

- diagonals bisect angles

- diagonals perpendicular

*Trapezoid I have only one set of parallel sides.

*Rectangle I have all of the properties of the parallelogram PLUS

- 4 right angles

- diagonals congruent

*Square Hey, look at me!

I have all of the properties of the parallelogram

AND the rectangle AND the rhombus.

I have it all!

*Isosceles Trapezoid I have:

- only one set of parallel sides

- base angles congruent

- legs congruent

- diagonals congruent

- opposite angles supplementary

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8.1 NOTES

Essential Vocabulary

Polygon

Not Polygons

Concave Convex

Equilateral Equiangular Regular

Triangle Octagon

Quadrilateral

Nonagon

Pentagon Decagon

Hexagon Dodecagon

Heptagon n-gon (an n-sided shape) ANY shape can be called “n”-gon based on the number of sides. Polygons with more than 10 sides are usually referred to as “n”-gons Ex. 14-gon, 32-gon, 100-gon

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Interior and Exterior Angles of a Polygon

In a polygon, two vertices that are endpoints of the same side are called consecutive

vertices. A diagonal of a polygon joins two non-consecutive vertices of a polygon.

Notice that when you draw all the diagonals of a polygon from one vertex, you divide the

polygon into ___________________________________. Recall that the triangle sum theorem states

_____________________________________________________________________________________________________.

For each polygon in the table, draw all the diagonals from one vertex. Complete the table.

Polygon # of Sides # of triangles

formed Interior Angle

Sum of Polygon

Triangle

Quadrilateral

Pentagon

Hexagon

Heptagon

n-gon

Consecutive vertices

Choose any vertex and

draw every diagonal

possible from that vertex.

POLYGON INTERIOR ANGLES THEOREM:

For an n-sided convex polygon, the sum of all the interior angles is ____________________.

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The exterior angle sum of a polygon does not depend on the number of sides on the

polygon. To prove this, use the diagrams below to calculate each exterior angle of the

polygons. Remember that an interior angle and its adjacent exterior angle form a linear

pair (so their sum is ___________).

triangle quadrilateral pentagon

85⁰ 110⁰ 130⁰

50⁰

125⁰ 115⁰ 85⁰

55⁰

90⁰ 40⁰ 95⁰ 100⁰

Interior angle sum

180⁰ 360⁰ 540⁰

Exterior angle sum

POLYGON EXTERIOR ANGLES THEOREM: The sum of all the exterior angles of a convex polygon (one exterior angle at each vertex is always ________________________.

Example 1: Find the sum of the measures of the interior

angles of a convex octagon.

Example 2: The sum of the measures of the interior angles

of a convex polygon is 1440⁰. How many sides

does the polygon have?

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Example 3: Find the value of x.

Example 5: Find the value of x.

Example 4: A trampoline is shaped like a regular

dodecagon (12 sides). Find the measure of

each interior and exterior angle.

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8.1 HOMEWORK

Find the sum of the measures of the interior angles of the indicated convex polygon.

1. 11-gon 2. 40-gon

The sum of the measures of the interior angles of a convex polygon is given. Classify the polygon by the number of sides.

3.

4.

5.

Find the value of . 6. 7.

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8. The measures of the interior angles of a convex octagon are and What is the measure of the smallest interior angle?

Find the measures of an interior angle and an exterior angle of the indicated polygon.

9. Regular Octagon

10. Regular 100-gon

11. The side view of a storage shed is shown below. Find the value of . Then determine the measure of each angle.

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8.2 NOTES

Properties of Parallelograms

DEFINITION: A parallelogram is a quadrilateral with __________________________________

_____________________________________________________________________________.

Other properties: If a quadrilateral is a parallelogram, then….

THM 8.3

THM 8.4

THM 8.5

THM 8.6

Ex1: Find the values of x and y.

Ex 2:

You can call this figure: “Parallelogram PQRS” or.

In , and , by definition.

Your thoughts:

1) Is it a parallelogram? YES, b/c the opp. sides

are parallel (def.)

2) Opposite sides are congruent, so x + 4 = 12.

3) Opposite angles are congruent, so y = 65.

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Ex 3-6:

Ex 7: The measure of one interior angle of a parallelogram is 50 degrees more than 4

times the measure of another angle. Find the measure of each angle.

(Make a sketch and label it.)

Ex 8: In LMNO , the ratio of LM to MN is 4:3. Find LM if the perimeter of LMNO is 28.

Ex 9: The diagonals of LMNO intersect at point P. What are the coordinates of P?

Hint: What do you need to know?

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Ex 10: Is the quadrilateral formed by the lines on the graph a parallelogram?

Hint: What do you need to know?

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8.2 HOMEWORK

Find the measure of the indicated angle in the parallelogram.

1. Find 2. Find

Find the value of each variable in parallelogram.

3. 4.

Find the indicated measure in

5.

6.

7.

8.

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Use the diagram of Points and are midpoints of and

Find the indicated measure.

9.

10.

11.

12.

13. In parallelogram the ratio of to is . Find if the perimeter of

is .

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8.3 NOTES

Proving a Quadrilateral is a Parallelogram

Examples:

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8.3 HOMEWORK

What theorem can you use to show that the quadrilateral is a parallelogram?

1. 2.

3. 4.

For what value of is the quadrilateral a parallelogram?

5. 6.

7. 8.

9. 10.

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The vertices of quadrilateral are given. Draw in a coordinate plane

and show that it is a parallelogram.

11. ( ) ( ) ( ) ( )

12. ( ) ( ) ( ) ( )

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8.4NOTES

Rhombuses, Squares and Rectangles

In this lesson, you will learn about three special types of parallelograms:

Rhombus Rectangle Square A parallelogram with four congruent sides (equilateral).

A parallelogram with four right angles (equiangular).

A parallelogram with four congruent sides and four right angles (regular).

SPECIAL NOTES ABOUT SQUARES:

Since a square has four congruent sides, it is also a _____________________________________ .

Since a square has four right angles, it is also a __________________________________________ .

Diagonals of Rhombuses and Rectangles

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Examples: Name each quadrilateral—parallelogram, rectangle, rhombus, and square—for which the statement is true.

1. It is equiangular.

2. It is equiangular and equilateral.

3. It is diagonals are perpendicular.

4. Opposite sides are congruent.

5. The diagonals bisect each other.

6. The diagonals bisect opposite angles. Classify the special quadrilateral. Explain your reasoning. Then find the values of and .

7. 8.

9. In the rhombus to the right, given . Find all the other angles. __________

__________

__________

__________

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10. Given rectangle CERT and , find each measure.

__________

__________

__________

__________

11. Given rectangle CERT,

(1) If and , solve for .

(2) If and , solve for .

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8.4 HOMEWORK For any rhombus decide whether the statement is always or sometimes true.

Draw a diagram and explain your reasoning.

1. 2.

For any rectangle decide whether the statement is always or sometimes true.

Draw a diagram and explain your reasoning.

3. 4.

Classify the quadrilateral. Explain your reasoning.

5. 6.

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Classify the special quadrilateral. Explain your reasoning. Then find the values of

and

7. 8.

The diagonals of rhombus intersect at Given that and find the indicated measure.

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10.

11.

12.

13. In preparation for a storm, a window is protected by nailing boards along its

diagonals. The lengths of the boards are the same. Can you conclude that the

window is a square? Explain.

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8.5/8.6 NOTES

Trapezoids & Kites

DEFINITION: A trapezoid is a quadrilateral with ________________________________________

_________________________________________________________________________________.

The parallel sides are called the __________________________________.

The non-parallel sides are called the __________________________________.

Since a trapezoid has two bases, it has two pairs of __________________________________.

DEFINITION: An isosceles trapezoid is one in which the __________________________________.

Think of it as an isosceles triangle with the “top” cut off by a segment parallel to a

base.

*An isosceles trapezoid has:

___________________________________________________

___________________________________________________

(THM 8.14)

(THM 8.16)

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DEFINITION: The midsegment of a trapezoid is a segment that connects the

____________________________ of its ____________________________.

(THM 8.17) The midsegment of a trapezoid is parallel to each base and measures on

half the sum of the base lengths.

**In other words,

Examples:

1. Find and . 2. Find the length of the midsegment.

3. Solve for . 4. In trapezoid , and . The midsegment is .

Sketch and its midsegment and find .

If MN is the midsegment of trapezoid 𝐴𝐵𝐶𝐷,

then MN AB, MN DC, and 𝑀𝑁 1

2(𝐴𝐵 𝐶𝐷)

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DEFINITION: A kite is a quadrilateral with ____________________________________________________.

THREE RULES FOR KITES

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UNIT 8 PERFORMANCE TASK

Performance of a Lifetime You are a dance choreographer and have been asked by the Houston Rockets to come up with a 45 second dance that will be showcased during a playoff game. You have been allowed to use the entire court for the performance and have been asked to make sure that all sides of the viewing audience will be able to see the performance without using the Jumbotron. Your goal is to fill the 94 x 50 foot court by placing the dancers in the shape of a polygon. You want to make sure that you use as much of the court as possible while making sure that there is an equal amount of unused space at each corner of the court.

1. Determine where the vertices of your polygon must fall to develop an equal amount of

unused space so all fans can see the performance and the owner of the Houston Rockets will be happy. How would you find these points and how do you know the unused space is equal at each corner of the court?

2. What type of polygon is created? How can you justify your answer? Justify all of the polygon’s properties.

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